Curvature scaling analysis of transient convective boundary layer flow induced by constant flux heating around a vertical cylinder for Prandtl number larger than one fluids

In this study, a scale analysis method is employed to investigate the time-evolving convective boundary layer flow on the external surface of a heated vertical cylinder under iso-flux conditions at Pr > 1. A series of numerical simulations are conducted over a wide range of flow parameters, including cylinder aspect ratios A, Rayleigh number Ra, and Prandtl number Pr. The aspect ratio A ranges from 1 to 100, while the Rayleigh number Ra spans from 1 × 107 to 1 × 109, and the Prandtl number Pr varies from 10 to 100. The present simulations indicate that, similar to the extensively studied flat-plate boundary layer flows, the development of the curved boundary layer also comprises three distinct states: an initial growth state, a transitional state, and a steady state. The results demonstrate that both the thickness and velocity of the boundary layer flow are significantly influenced by the curvature effect, with the maximum velocity of the boundary layer in the initial state being particularly sensitive to the curvature effect. Specifically, when A = 100, this maximum velocity can be reduced by up to 60%. Explicit scale laws are proposed and they could accurately describe these influence for both initial growth and steady states. These scaling laws consist of the flat plate scaling, which accounts for the dependencies on Ra and Pr, and a dimensionless explicit curvature coefficient Ψn(A), which describes the curvature effect dependencies. When the curvature of the heated wall is minimal, the proposed scaling law converges to the well-established flat boundary layer scaling laws, with the curvature coefficient Ψn(A) approaching the unity. Conversely, when the curvature of the heated wall is substantial, the curvature coefficient Ψn(A) provides an accurate quantification of this effect. A comparison between the scaling laws and numerical calculations reveals that most fitting constants R2 are approximately 0.999, thereby providing strong validation for the proposed scaling laws.